Preheat the oven to 350°F (175°C). Butter and flour a 10-inch round, 2-inch-deep springform pan and line the bottom with parchment paper. Wrap the outside of the pan with several layers of aluminum foil to prevent leaks. Also line a rimmed baking sheet with aluminum foil.
In a large bowl with a stand mixer or a hand-held mixer, beat 1 1/2 cups sugar and the butter until very light in color, 3 to 5 minutes. Scrape the bowl as needed with a rubber spatula and continue to beat until the mixture becomes very light in texture, 3 to 4 minutes more.
Beat in the eggs, 1 at a time, scraping the bowl between additions.
Sift together the flours, baking powder, and salt. Alternately add the milk and dry ingredients to the butter mixture, stopping to scrape the bowl as necessary. Add the vanilla and mix the batter just until smooth. Do not overbeat the batter.
Pour the batter into the prepared pan and spread it evenly with a rubber spatula. Place the apple slices on the batter, overlapping one another in concentric circles and completely covering the surface of the cake. Repeat layering the apples in concentric circles until all of them are used or you reach almost the top of the pan, whichever happens first. Pour the cream evenly over the apples. Stir together the remaining 1/2 cup sugar and the cinnamon and sprinkle over the top of the cake.
Place the pan on the prepared baking sheet and bake for anywhere from 55 to 75 minutes, until a toothpick inserted in the center comes out clean. Remove the pan from the oven, transfer it to a wire rack, and let the cake cool at least 15 minutes. Gently remove the sides of the pan and let the cake continue to cool to room temperature.
Invert the cake onto the rack and remove the pan and the parchment paper. Then invert it once more onto a serving platter. Serve the apple cake warm or room temperature. (The cooled apple cake can be stored tightly covered in the refrigerator for up to 3 days.) Originally published September 23, 2003.